Sierpiński space

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Not to be confused with Sierpiński set.

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

The Sierpiński space has important relations to the theory of computation and semantics.[1][2]

Definition and fundamental properties

Explicitly, the Sierpiński space is a topological space S whose underlying point set is {0,1} and whose open sets are

$\{\varnothing,\{1\},\{0,1\}\}.$

The closed sets are

$\{\varnothing,\{0\},\{0,1\}\}.$

So the singleton set {0} is closed (but not open) and the set {1} is open (but not closed).

The closure operator on S is determined by

$\overline{\{0\}} = \{0\},\qquad\overline{\{1\}} = \{0,1\}.$

A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by

$0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.$

Topological properties

The Sierpiński space S is a special case of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore S has many properties in common with one or both of these families.

Compactness

• Like all finite topological spaces, the Sierpiński space is both compact and second-countable.
• The compact subset {1} of S is not closed showing that compact subsets of T0 spaces need not be closed.
• Every open cover of S must contain S itself since S is the only open neighborhood of 0. Therefore every open cover of S has an open subcover consisting of a single set: {S}.
• It follows that S is fully normal.[3]

Convergence

• Every sequence in S converges to the point 0. This is because the only neighborhood of 0 is S itself.
• A sequence in S converges to 1 if and only if the sequence contains only finitely many terms equal to 0 (i.e. the sequence is eventually just 1's).
• The point 1 is a cluster point of a sequence in S if and only if the sequence contains infinitely many 1's.
• Examples:
• 1 is not a cluster point of (0,0,0,0,…).
• 1 is a cluster point (but not a limit) of (0,1,0,1,0,1,…).
• The sequence (1,1,1,1,…) converges to both 0 and 1.

Continuous functions to the Sierpiński space

Let X be an arbitrary set. The set of all functions from X to the set {0,1} is typically denoted 2X. These functions are precisely the characteristic functions of X. Each such function is of the form

$\chi_U(x) = \begin{cases}1 & x \in U \\ 0 & x \not\in U\end{cases}$

where U is a subset of X. In other words, the set of functions 2X is in bijective correspondence with P(X), the power set of X. Every subset U of X has its characteristic function χU and every function from X to {0,1} is of this form.

Now suppose X is a topological space and let {0,1} have the Sierpiński topology. Then a function χU : XS is continuous if and only if χU−1(1) is open in X. But, by definition

$\chi_U^{-1}(1) = U.$

So χU is continuous if and only if U is open in X. Let C(X,S) denote the set of all continuous maps from X to S and let T(X) denote the topology of X (i.e. the family of all open sets). Then we have a bijection from T(X) to C(X,S) which sends the open set U to χU.

$C(X,S)\cong \mathcal{T}(X)$

That is, if we identify 2X with P(X), the subset of continuous maps C(X,S) ⊂ 2X is precisely the topology of X: T(X) ⊂ P(X).

Categorical description

The above construction can be described nicely using the language of category theory. There is contravariant functor T : TopSet from the category of topological spaces to the category of sets which assigns each topological space X its set of open sets T(X) and each continuous function f : XY the preimage map

$f^{-1} : \mathcal{T}(Y) \to \mathcal{T}(X).$

The statement then becomes: the functor T is represented by (S, {1}) where S is the Sierpiński space. That is, T is naturally isomorphic to the Hom functor Hom(–, S) with the natural isomorphism determined by the universal element {1} ∈ T(S).

The initial topology

Any topological space X has the initial topology induced by the family C(X,S) of continuous functions to Sierpiński space. Indeed, in order to coarsen the topology on X one must remove open sets. But removing the open set U would render χU discontinuous. So X has the coarsest topology for which each function in C(X,S) is continuous.

The family of functions C(X,S) separates points in X if and only if X is a T0 space. Two points x and y will be separated by the function χU if and only if the open set U contains precisely one of the two points. This is exactly what it means for x and y to be topologically distinguishable.

Therefore if X is T0, we can embed X as a subspace of a product of Sierpiński spaces, where there is one copy of S for each open set U in X. The embedding map

$e : X \to \prod_{U\in \mathcal{T}(X)}S = S^{\mathcal{T}(X)}$

is given by

$e(x)_U = \chi_U(x).\,$

Since subspaces and products of T0 spaces are T0, it follows that a topological space is T0 if and only if it is homeomorphic to a subspace of a power of S.

In algebraic geometry

In algebraic geometry the Sierpiński space arises as the spectrum, Spec(R), of a discrete valuation ring R such as Z(2) (the localization of the integers at the prime ideal generated by 2). The generic point of Spec(R), coming from the zero ideal, corresponds to the open point 1, while the special point of Spec(R), coming from the unique maximal ideal, corresponds to the closed point 0.